TY  - JOUR
A1  - Harutyunyan, Gohar
A1  - Schulze, Bert-Wolfgang
T1  - The Zaremba problem with singular interfaces as a corner boundary value problem
JF  - Potential analysis : an international journal devoted to the interactions between potential theory, probability theory, geometry and functional analysis
N2  - We study mixed boundary value problems for an elliptic operator A on a manifold X with boundary Y, i.e., Au = f in int X, T (+/-) u = g(+/-) on int Y+/-, where Y is subdivided into subsets Y+/- with an interface Z and boundary conditions T+/- on Y+/- that are Shapiro-Lopatinskij elliptic up to Z from the respective sides. We assume that Z subset of Y is a manifold with conical singularity v. As an example we consider the Zaremba problem, where A is the Laplacian and T- Dirichlet, T+ Neumann conditions. The problem is treated as a corner boundary value problem near v which is the new point and the main difficulty in this paper. Outside v the problem belongs to the edge calculus as is shown in Bull. Sci. Math. ( to appear). With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along Z {v} of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions.
KW  - Zaremba problem
KW  - corner Sobolev spaces with double weights
KW  - pseudo-differential boundary value problems
Y1  - 2006
U6  - https://doi.org/10.1007/s11118-006-9020-6
SN  - 0926-2601
VL  - 25
SP  - 327
EP  - 369
PB  - Springer
CY  - Dordrecht
ER  - 
TY  - INPR
A1  - Kapanadze, David
A1  - Schulze, Bert-Wolfgang
T1  - Symbolic calculus for boundary value problems on manifolds with edges
N2  - Boundary value problems for (pseudo-) differential operators on a manifold with edges can be characterised by a hierarchy of symbols. The symbol structure is responsible or ellipicity and for the nature of parametrices within an algebra of "edge-degenerate" pseudo-differential operators. The edge symbol component of that hierarchy takes values in boundary value problems on an infinite model cone, with edge variables and covariables as parameters. Edge symbols play a crucial role in this theory, in particular, the contribution with holomorphic operatot-valued Mellin symbols. We establish a calculus in s framework of "twisted homogenity" that refers to strongly continuous groups of isomorphisms on weighted cone Sobolev spaces. We then derive an equivalent representation with a particularly transparent composition behaviour.
T3  - Preprint - (2001) 21 
KW  - pseudo-differential boundary value problems
KW  - operators on manifolds with singularities
Y1  - 2001
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26046
ER  - 
TY  - INPR
A1  - Kapanadze, David
A1  - Schulze, Bert-Wolfgang
T1  - Pseudo-differential crack theory
N2  - Crack problems are regarded as elements in a pseudo-differential algbra, where the two sdes int S± of the crack S are treated as interior boundaries and the boundary Y of the crack as an edge singularity. We employ the pseudo-differential calculus of boundary value problems with the transmission property near int S± and the edge pseudo-differential calculus (in a variant with Douglis-Nirenberg orders) to construct parametrices od elliptic crack problems (with extra trace and potential conditions along Y) and to characterise asymptotics of solutions near Y (expressed in the framework of continuous asymptotics). Our operator algebra with boundary and edge symbols contains new weight and order conventions that are necessary also for the more general calculus on manifolds with boundary and edges.
T3  - Preprint - (2000) 09 
KW  - Crack theory
KW  - pseudo-differential boundary value problems
KW  - operator algebras on manifolds with singularities
KW  - conormal asymptotics
Y1  - 2000
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25759
ER  -